$\iint_D \frac{1}{1+x^2+y^2} dx dy$

Question

Problem: $$\iint_D \frac{1}{1+x^2+y^2} dx dy,$$ where $D=\left\{(x,y):0 \le x \le y\right\}$.

I got everything right, except the region. The book said $\pi / 4 \le \theta \le \pi /2$ and I wanted the region where $0 \le \theta \le \pi /4 $. Since I wrote up that $D$ was the region where $0 \le y$ and $0 \le x \le y$.

Answer 1

$x\le y$ means $r\cos\theta\le r\sin\theta$, or (assuming they are positive), $\tan\theta\ge 1$, so ${\pi \over 4}\le\theta\le{\pi\over 2}$.